Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{-7k^3 + 7k}{k^3 - 2k^2 - 3k}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {-7k(k^2 - 1)} {k(k^2 - 2k - 3)} $ $ p = -\dfrac{7k}{k} \cdot \dfrac{k^2 - 1}{k^2 - 2k - 3} $ Simplify: $ p = - 7 \cdot \dfrac{k^2 - 1}{k^2 - 2k - 3}$ Since we are dividing by $k$ , we must remember that $k \neq 0$ Next factor the numerator and denominator. $ p = - 7 \cdot \dfrac{(k + 1)(k - 1)}{(k + 1)(k - 3)}$ Assuming $k \neq -1$ , we can cancel the $k + 1$ $ p = - 7 \cdot \dfrac{k - 1}{k - 3}$ Therefore: $ p = \dfrac{ -7(k - 1)}{ k - 3 }$, $k \neq -1$, $k \neq 0$